Heteroclinic orbits indicate overexploitation in predator prey systems with a strong Allee effect
Voorn, G. A. K. van, Hemerik, L., Boer, M. and Kooi, B. W.
2007. Heteroclinic orbits indicate overexploitation in predator prey systems with a strong Allee effect.
Math Biosci 209: 451 - 469
Abstract
Successful prey species establishment in a model system can be
dependent not only on the parameter setting, but also on the initial
conditions of the system. Similarly, successful predator species
establishment can be dependent on both these conditions. Predator
invasion can fail and lead to system collapse, an event referred to
as overexploitation. This phenomenon is especially important in
models with bistability properties, such as strong Allee effects. The
prey extinction threshold then prevents easy re-establishment of the
prey species. In this paper we deal with the bifurcation analyses of
two previously published predator prey models with strong Allee
effects. We expand the analyses to include not only local, but also
global bifurcations. We show the existence of a point-to-point
heteroclinic cycle in these models, and discuss numerical techniques
for continuation in parameter space. The continuation of such a
cycle in two-parameter space forms the boundary of a region in
parameter space where the system collapses after predator invasion,
i.e. where overexploitation occurs. We argue that the detection and
continuation of global bifurcations in these models are of vital
importance for the understanding of the model dynamics.
Keywords: Bistability, depensation, extinction threshold,
global bifurcation, heteroclinic point-to-point connection,
separatrix